Spectral and Scattering Theory and its Application

本文主要研究了薛定谔算子的谱和散射理论中的三个问题：(1)对于指数乘积公式(Lie-Trotter-Kato乘积公式)，证明了算子范数收敛，并建立了误差估计。所得结果已应用于具有奇异势或含时变势的Schr_dinger半群或传播子。(2)无电势的未受扰动的Pauli算子的本征值为零且有无穷多个本征值作为其本质谱的底。当算子被无穷大处的势扰动时，研究了离散本征值在原点附近的渐近分布。特别强调了Pauli算符不一定具有恒定磁场的情况。(3)分析了二维磁场散射在低能散射幅度下的渐近行为，并讨论了与小支撑磁场散射的关系。

The present project has been devoted to the study on the following three subjects related to the spectral and scattering theory for Schr_dinger operators.(1) For exponential product formula(Lie-Trotter-Kato product formula), the convergence in operator norm has been proved and the error estimate has been also established. The 9btained results have been applied to Schr_dinger semi-groups or propagators with singular or time-dependent potentials.(2) The unperturbed Pauli operator without electric potentials has zero eigenvalue with infinite multiplicities as its bottom of essential spectrum. When the operators are perturbed by potentials falling off at infinity, the asymptotic distribution of discrete eigenvalues near the origin has been studied. The special emphasis is placed on the case that Pauli operators do not necessarily have constant magnetic fields.(3) The asymptotic behavior at low energy of scattering amplitudes has been analysed for scattering by two dimensional magnetic fields and the relation to scattering by magnetic fields with small support has been also discussed.